Looking at bitcoinwatch.com there is a line that tells you how many bitcoins were sent in the last 24 hours. At the moment it is a whole 8% of the total bitcoins in existence.

This makes the bitcoin market appear to be very lively, if 8% of all bitcoins exchange hands on a daily basis.

However, as I understand it, when you send "n" amount of bitcoins from your wallet, all of your bitcoins "x" are sent from your wallet, of which "x-n" are sent to a new adders back you your wallet, and "n" bitcoins are sent to the address you specified. So is your wallet has 100 BTC, and you send 1 BTC, the network sees it as 100 BTC being sent. 99 BTC to you, 1 BTC to the recipient.

Is that how the stat works? Or does it not count the "x-n" BTC?

Also, lets say you send 1 BTC to one person, and 1 BTC to another person, does the stat count it as 100 BTC sent and then another 99 BTC sent, for a total of 199?

It's not ALL of your bitcoin that get sent. It's all of the bitcoins from all the addresses that will be needed to compose the value you wish to send. The remaining value from those inputs will be sent back to you.

That isn't how bitcoin works. Bitcoin can't spend part of an OUTPUT. So all tx must include entire OUTPUT(s) not entire wallets. So if you have 100 BTC in your wallet, you want to send 10 BTC, and the smallest output (or sum of outputs) which will result in no fee is 12 BTC then you send 12 (100), 10 to the destination and 2 back to a change address.

Still what makes calculating tx value difficulty is that it is never possible to know for sure which is the "payment". In the example above you could be sending 10 (w/2 change) or sending 2 (w/ 10 change).

BlockChain.info attempts to make educated guesses to determine which was the intended tx.

How should I interpret bitcoin days destroyed? It seems if velocity is low then the number of days goes up. If there are no transactions then this number is zero.

If you spend 1 BTC which is 100 days old then it is 100 bitcoin days destroyed. Some incorrectly dismiss this because moving some very old coins can result in a lot of days destroyed HOWEVER they are now 0 day coins. Each individual move isn't important what is interesting is the aggregate daily # of coins destroyed and if that amount is rising or falling.